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For example, if I had a random variable X and knew it followed a normal distribution, I might write:

$X \sim N(5, 10)$

What would I write if I'm trying to say that the distribution is unknown?

$X \in \Omega$

Is that correct? It doesn't feel right.

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    $\begingroup$ If you have previously defined or described $\Omega$ as a set of distributions that would be perfectly fine and clear. Because conventionally "$\Omega$" is already used to refer to a sample space, though, that might confuse an experienced statistical audience. $\endgroup$
    – whuber
    Commented Feb 3, 2023 at 13:14
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    $\begingroup$ I would just say that $X$ is a random variable. $\endgroup$
    – Valentas
    Commented Feb 3, 2023 at 21:25
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    $\begingroup$ Unless there's some special reason you need to use mathematical notation, I would suggest to use narration instead: "Let $X$ be a random variable with [density $\delta$/cdf $F$/positive support/...]". $\endgroup$ Commented Feb 4, 2023 at 20:59

2 Answers 2

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The notation I tend to see is something like $X\sim F_X$ to denote that $X$ is a random variable with $F_X$ as its CDF. I have seen people try to be brief and just write $X\sim F$, but this could mislead others into thinking that $X$ has an $F$ distribution, when that could be far from the case.

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  • $\begingroup$ Just to nitpick, we have books that go for $X\sim P_X$ too. $\endgroup$ Commented Feb 3, 2023 at 23:22
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Standard notation is $X\sim F$ or $X\sim F(x).$.

Update: the latter notation, while common shorthand, could be misunderstood since $F(x)$ is a probability.

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    $\begingroup$ I can't explain the downvote, but it made me look closely at this answer and I can't help noticing that "$X\sim F(x)$" is a mathematical solecism: $F$ might be a distribution function, but then $x$ must be a possible value of $X$ and therefore "$F(x)$" refers to a probability -- a single number. $\endgroup$
    – whuber
    Commented Feb 3, 2023 at 19:46
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    $\begingroup$ My, the nitpicking is strong with this community. $\endgroup$ Commented Feb 3, 2023 at 22:12
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    $\begingroup$ @Stephan Nitpicking maybe--but that kind of abuse of notation is the source of plenty of misconceptions IMHO. $\endgroup$
    – whuber
    Commented Feb 3, 2023 at 23:50
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    $\begingroup$ @whuber that's why I started with F , and only as second option F(x). You could also write X~F(y). Hence why included it for clarity ( even though technically it's a probability). I doubt anyone who is familiar with probability/stats to the degree of sample spaces has a misconception (!) By F vs F(x) notation. I ll be more precise next time $\endgroup$ Commented Feb 4, 2023 at 3:04
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    $\begingroup$ @whuber: I'm all with you on this, I have tried to get "nitpicker in chief" as a job title. I just found the five upvotes to your comment versus the zero net votes on the answer a trifle remarkable... $\endgroup$ Commented Feb 4, 2023 at 6:14

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